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	\begin{table}[htbp] 
  \centering
  \caption{\bf Sharpe ratios for local-PCA ($S \& P \, 500$):}
  \centering {\bf  In-Sample vs Out-of-Sample ($T=24$, Tangency - pa)}
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\begin{tabular}{lccccccccccc}
  \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    &   - &    0.621 &  0.395 &  0.375 &  0.325 &  0.413 &  0.296 & -0.316 &  0.326 & -0.331 & -0.335 \\
  local-PCA (bal) &    0.169 & 0.862 &  0.265 & 0.338 & 0.096 & 0.253 & 0.360 & 0.498 &  0.545 & 0.388 & 0.417 \\
local-PCA (unbal) &  
 0.744 & 0.843 &  0.708 & 0.591& 0.588& 0.591& 0.722 & 0.649 & 0.962 & 0.932 & 0.804  \\ \hline
   \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    & -  &  
  0.430 & 0.413 & 0.414 & 0.579 & 0.578 & 0.579 & 0.580 & 0.611 &  0.611 & 0.627 \\
  local-PCA (bal) & 
   0.799  &  0.953 & 1.151 & 1.450 & 1.804 & 1.993 &  2.266 &  2.459 &  2.615&  2.880 &  3.099 \\
    local-PCA (unbal) &   
  1.146  & 0.758 & 0.819 & 0.954 & 1.05 & 1.273 & 1.558 & 1.74 &  2.012 &  2.121 &  2.210 \\ \hline
     \end{tabular}
        \parbox{6.5in}{\footnotesize{This  table presents the  time-average (across $636-2T$ windows) of the time series for the annualized Sharpe ratios  
         both in-sample (bottom panel) and out-of-sample (top panel) for the Tangency Portfolio,
    correspondingly to  the (time-varying) estimated number of factors $\tilde{r}$   (column two), described in Section~OA15.1.1 and to the values one to  $10 $ (columns 3-12), respectively.  The local-PCA is evaluated with $T=24$. Data are monthly from Dec 1960 until Dec 2013.}}
 \label{Table8OASP500}
\end{table}

	
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